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A166509 Decimal expansion of Sum_{p prime} (Kronecker(-1/p)/p). 3
1, 6, 5, 0, 1, 8, 6, 7, 4, 7, 0, 0, 0, 0, 6, 8, 1, 8, 9, 3, 6, 6, 8, 2, 8, 7, 8, 5, 1, 2, 4, 5, 6, 4, 2, 6, 2, 2, 0, 0, 2, 4, 6, 1, 9, 2, 4, 4, 9, 2, 2, 9, 5, 1, 8, 9, 1, 9, 7, 9, 4, 2, 1, 1, 5, 4, 7, 7, 7, 1, 5, 6, 7, 2, 8, 1, 1, 5, 8, 8, 9, 3, 7, 5, 1, 0, 0, 3, 6, 8, 9, 7, 0, 1, 9, 6, 6, 5, 4, 6, 0, 7, 5, 1, 3 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

As a decimal number, this equals 0.5 - A086239. The Kronecker-Legendre-Jacobi symbol Kronecker(-1/p) equals -1 if p=3 (mod 4) and +1 for all other primes. The first ten digits have been computed using the numerical result from Finch for A086239.

Using results from Cohen's preprint, David Broadhurst computed 300 decimals in less than 0.2 seconds (cf. link to primenumbers group). He reports that the method can give 70 correct decimals using only two odd primes, p=3 and p=5. - M. F. Hasler, Oct 29 2009

Contrary to an earlier comment, this constant is not hard to evaluate: see the link for the first 10000 decimal digits. - David Broadhurst, Oct 30 2009

REFERENCES

S. R. Finch, Meissel-Mertens Constants, Sect. 2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, 2003, p. 96.

LINKS

Table of n, a(n) for n=0..104.

J. P. Benney and others, Is this a convergent series and if so what is its sum?, in primenumbers group, Oct 26 2009.

David Broadhurst, 10000 digits [From David Broadhurst, Oct 30 2009]

David Broadhurst, 10000 digits [Cached copy]

Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, preprint (1991). [M. F. Hasler, Oct 29 2009]

Eric Weisstein's World of Mathematics, Prime Sums

EXAMPLE

0.1650186747... = 1/2 - 1/3 + 1/5 - 1/7 - 1/11 + 1/13 + 1/17 ... with negative sign iff p=3 (mod 4).

PROG

(PARI) p=1; sum( i=2, 10^6, kronecker(-1, p=nextprime(p+2))/p, .5) /* the first 10^6 terms give 0.16502... */

(PARI) /* The given parameters NP, N yield the correct result for default(realprecision, 300) in less than 0.5 seconds. Code based on David Broadhurst's script posted on primenumbers group. */

A166509(NP=70, N=115)={ local( P=vector(NP, k, (2-prime(k)%4)/prime(k)));

sum( s=1, N, moebius(s)/s*log( prod( k=2, #P, 1-P[k]^s,

if( s%2, if( s==1, Pi/4, sumalt( k=0, (-1)^k/(2*k+1)^s)), zeta(s)*(1-1/2^s)) /* end if s%2 */

)), sum(k=2, #P, P[k], 1/2.))} \\ M. F. Hasler, Oct 29 2009

CROSSREFS

Cf. A166510.

Sequence in context: A264807 A245973 A214128 * A200635 A197261 A010773

Adjacent sequences:  A166506 A166507 A166508 * A166510 A166511 A166512

KEYWORD

cons,easy,nonn

AUTHOR

M. F. Hasler, Oct 28 2009

EXTENSIONS

More terms, computed using David Broadhurst's method, from M. F. Hasler, Oct 29 2009

STATUS

approved

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Last modified October 18 08:04 EDT 2019. Contains 328146 sequences. (Running on oeis4.)